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Homework Assignment 11 in Differential Equations, MATH308-FALL 2015 Total number of points: 150 due November 4, 2015 Sections covered: end of 7.5 (the case when there are repeated eigenvalues and a basis of eigenvectors)& 7.8 (when there are repeated eigenvalues but no basis of eigenvectors) 1. Given the following system of linear differential equations: 0 x1 = x2 − 4x3 x0 = −9x1 + 6x2 − 12x3 20 x3 = −3x1 + x2 − x3 (a) Find the eigenvalues of the corresponding matrix and for each eigenvalue determine its algebraic and geometric multiplicities; (b) Find the general solution. 2. Given the following system of linear differential equations: 0 x1 = x1 − 29 x2 x02 = 2x1 + 7x2 (1) (a) Find the general solution of the system (1). x1 (t) (b) If x(t) = is a solution of (1), what is the limit of x(t) as t → −∞. Does this limit x2 (t) depend on initial conditions? (c) Find the solution of the system (1) satisfying the initial conditions: x1 (0) = 3, x2 (0) = −4. 3. Given the following system of linear differential equations: 0 x1 = 5x + 4x2 − 5x3 x0 = −16x1 − 4x2 + 15x3 20 x3 = 2x1 + 4x2 − 2x3 (a) Find the eigenvalues of the corresponding matrix and for each eigenvalue determine its algebraic and geometric multiplicities; (b) Find the general solution. 1